MATH 120-Linear Algebra With Differential Equations-Muhammad Ahsan.pdf

Lahore University of Management Sciences

MATH 120 – Linear Algebra with Differential Differential Equations Fall 2015-2016

Instructor Room No. Office Hours Email Telephone Secretary/TA TA Office Hours Course URL (if any)

Muhammad Ahsan 9-245 TBA

[email protected]  8011 TBA TBA math.lums.edu.pk/moodle

Course Basics Credit Hours Lecture(s) Recitation/Lab (per week) Tutorial (per week)

3 Nbr of Lec(s) Per Week Nbr of Lec(s) Per Week Nbr of Lec(s) Per Week

Course Distribution Core Elective Open for Student Category Close for Student Category

All students None

2

Duration Duration Duration

75min

COURSE DESCRIPTION This is the first course of a two semester sequence in linear algebra. This course gives a working knowledge of: systems of linear equations, matrix algebra, determinants, eigenvectors and eigenvalues, finite-dimensional vector spaces, matrix representations representations of linear transformations, matrix diagonalization, changes of basis, Separable and first-order linear equations with applications, 2nd order linear equations with

constant coefficients, method of undetermined coefficients, Systems of linear ODE's with constant coefficients, Solution by eigenvalue/eigenvectors, Non homogeneous linear systems . COURSE Anti-PREREQUISITE(S) Anti-PREREQUISITE(S) 

Math in A-levels, FSc, or the equivalent

COURSE OBJECTIVES   

To acquire a good understanding of the c oncepts and methods of linear algebra To develop the ability to solve problems using the techniques of linear algebra To develop critical reasoning by writing short proofs based on the axiomatic method To compute the solution of first order and higher order Ordinary differential equations To solve system of linear ODEs using eigenvalues and eigenvectors

Learning Outcomes Students will learn to Set up and solve systems of linear equations Perform matrix operations as appropriate Evaluate determinants and use their properties Understand and use linear transformations   

Lahore University of Management Sciences Work in real vector spaces Use the concepts of subspace, basis, dimension, row space, column space, row rank, column rank, and nullity Use inner products Use and construct orthonormal bases Apply linear algebra for best approximation and least squares fitting Evaluate and apply eigenvectors and eigenvalues Understand the features of general linear t ransformations such as kernel, range, inverses, matrix representations, similarity , and isomorphism Solve first and higher order ODEs Solve system of linear ODEs using eigenvalues and eigenvectors Use Mathematica and Maple to solve ODEs and system of ODEs Grading Breakup and Policy Assignment(s): 10 % Home Work: Quiz(s): 20% Class Participation: Attendance: 0 Midterm Examination: 30% Project: Final Examination: 40%

Examination Detail

Midterm Exam

Yes/No: Yes Combine/Separate: Combine Duration: 90min Exam Specifications: No notes/No books/No calculators

Final Exam

Yes/No: Yes Combine : Duration: 180min Exam Specifications: No notes/No books/No calculators

COURSE OVERVIEW

Week/ Lecture/ Module Part (i)

Topics

Recommended Readings

Objectives/ Application

Systems of linear equations

Chapter 1

Systems of linear equations and matrices

Gaussian elimination

Chapter 1 Section 1.1 1.2 Chapter 1 Section 1.3

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Determinants

Chapter 2

Systems of linear equations and matrices Systems of linear equations and matrices Systems of linear equations and matrices Systems of linear equations and matrices Systems of linear equations and matrices Systems of linear equations and matrices Systems of linear equations and matrices Determinants

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Cofactor expansion

Section 2.1 and 2.2

Determinants

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Matrices and matrix operations Matrix arithmetic Inverses Elementary matrices and inverses Further results on systems of linear equations and inverses Diagonal, triangular, and symmetric matrices

Chapter 1 1.4 Chapter 1 1.4 Chapter 1 1.5 Chapter 1 1.6 Chapter 1 1.7

Section Section Section Section Section

Lahore University of Management Sciences Properties of determinants

Section 2.3

Determinants

Euclidean vector spaces

Chapter 4

Euclidean vector spaces

Euclidean n-space Linear transformations from Rm to Rn Linear transformations and polynomials

Section 4.1

Euclidean vector spaces

Section 4.2 and 4.3

Euclidean vector spaces

Section 4.4

Euclidean vector spaces

General Vector Space

Chapter 5 Section 5.1 Section 5.2

Vector spaces Vector spaces

Section 5.4

Vector spaces

Section 5.5

Vector spaces

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Real vector spaces Subspaces Basis and dimension Row space, column space, null space Rank and nullity

Section 5.6

Vector spaces

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Inner Product Spaces

Chapter 6

Inner product spaces

Section 6.1

Inner product spaces

Angle and orthogonality Orthonormal basis Gram-Schmidt process Change of basis

Section 6.2

Inner product spaces

Section 6.3

Inner product spaces

Section 6.3

Inner product spaces

Section 6.5

Inner product spaces

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Orthogonal matrices

Section 6.6

Inner product spaces

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Eigenvalues and eigenvectors

Chapter 7

Eigenvalues and eigenvectors Diagonalization Orthogonal diagonalization Ordinary differential equations

Section 7.1

Eigenvalues and eigenvectors

Section 7.2

Eigenvalues and eigenvectors

Section 7.3

Eigenvalues and eigenvectors

Introduction to differential equations

Chapter 1

Basic definitions and terminology First order differential equations

Sections 1.1, 1.2

Separable and first-order linear equations with applications,

Section 2.1,2.2, 2.3

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Differential equations of higher order

Chapter 4

Homogeneous equations, Non-homogeneous equation Higher order linear equations with constant coefficients

Section 4.1, 4.2

Differential equations of higher order

Section 4.3

Differential equations of higher order

Systems of linear first order differential equations

Chapter 8

Homogeneous linear systems with constant coefficients Solution by eigenvalue/eigenvectors, nonhomogenous linear systems

Section 8.1, 8.2

Systems of linear first order differential equations

Section 8.2, 8.3

Systems of linear first order differential equations

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Part (ii)

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Differential equations with boundary value problems by Dennis G Zill

Chapter 2 First order differential equations

Textbook(s)/Supplementary Readings There is no required text but the following texts will be used for reference. 1. 2.

Elementary linear algebra (2005) Howard Anton, 9th edition, John Wiley and Sons Differential equations with boundary-value problems by Dennis G. Zill and Michael R. Cullin (5t h Edition Brooks/Cole)

Handouts on topics will also been uploaded on the LUMS website

Lahore University of Management Sciences Helping Software’s : Mathematica Maple 14, 16

A first course in linear algebra, RA Beezer, http://linear.ups.edu/